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Well known theorems on triangular systems and the D 5
principle
François Boulier
François Lemaire
Marc Moreno Maza
Abstract
hal-00137158, version 1 - 22 Mar 2007
The theorems that we present in this paper are very important to prove the correctness of triangular decomposition algorithms. The most important of them are not new
but their proofs are. We illustrate how they articulate with the D5 principle.
Introduction
This paper presents the proofs of theorems which constitute the basis of the triangular
systems theory: the equidimensionality (or unmixedness) theorem for which we give two
formulations (Theorems 1.1 and 1.6) and Lazard’s lemma (Theorem 2.1). The first section
of this paper is devoted to the proof of the equidimensionality theorem. Our proof is original
since it covers in the same time the ideals generated by triangular systems saturated by
∞ ) and those saturated by
the set of the initials of the system (i.e. of the form (A) : IA
∞ ). The former type of
the set of the separants of the system (i.e. of the form (A) : SA
ideal naturally arises in polynomial problems while the latter one naturally arises in the
differential context. Our proof shows also the key role of Macaulay’s unmixedness theorem
[24, chapter VII, paragraph 8, Theorem 26]. Its importance in the context of triangular
systems was first demonstrated by Morrison in [14] and published in [15]. In her papers,
Morrison aimed at completing the proof of Lazard’s lemma provided in [3, Lemma 2]. Thus
∞ , which are the ideals
Morrison only considered the case of the ideals of the form (A) : SA
∞ was
w.r.t. which Lazard’s lemma applies. The case of the ideals of the form (A) : IA
addressed in [2]. The proof of [2, Theorem 5.1] involves the same gap as that given in [3,
Lemma 2]. It was fixed in [1]. The proof provided in [1] does not explicitly use Macaulay’s
theorem but relies on the properties of regular sequences in Cohen–Macaulay rings, which
are the rings in which Macaulay’s theorem applies.
What is this gap in the proofs mentioned above ? Among all the indeterminates the
elements of a triangular system A depend on, denote t1 , . . . , tm the ones which are not main
indeterminates. The proofs given in [2, Theorem 5.1] and in [3, Lemma 2] rely implicitly
on the assumption that the non zero polynomials which only depend on t1 , . . . , tm are not
zero divisors modulo the ideal defined by A. This assumption is indeed true but certainly
deserves a specific proof.
∞ , let’s mention the equidimensionality result
In the case of the ideals of the form (A) : IA
of [19] which is not sufficient since it does not solve the problem of the embedded associated
1
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∞ . In the case of the ideals of the form (A) : S ∞ , there is a simple proof
prime ideals of (A) : IA
A
∞.
[16, 4] which unfortunately does not seem to generalize to the ideals of the form (A) : IA
The second section of this paper is devoted to the proof of Lazard’s lemma. This lemma
was communicated by Lazard to the first author a few days before his PhD defense in
1994, with a sketch of proof. The proof given here is very close to the original one. As
stated above, Lazard’s lemma was first published in [3] but its first complete proof is due
to Morrison [14, 15]. Among the few other proofs published afterwards, let’s mention the
ones given in [18, 4, 8, 17].
In the remaining sections, we show how the equidimensionality theorem and Lazard’s
lemma apply to the so called “regular chains” [11, 9, 23, 2]. We last recall a few basic
algorithms which carry out a generalization of the “D 5 ” principle [6] for regular chains and
which implictly rely on the equidimensionality theorem. Historically, the “D 5 ” principle
suggests to compute modulo zero dimensional ideals presented by triangular systems as if
these ideals were prime (whenever a zero divisor is exhibited, the ideal is split). It is its
generalization to non zero dimensional ideals which requires the equidimensionality theorem.
Throughout this paper, K denotes a commutative field of characteristic zero.
1
The equidimensionality theorem
In the polynomial ring R = K[x1 , . . . , xn , t1 , . . . , tm ], we consider a polynomial system
A = {p1 , . . . , pn }. We assume that deg(pi , xi ) > 0 and deg(pi , xk ) = 0 for all 1 ≤ i ≤ n
and i < k ≤ n i.e. that A is a triangular system w.r.t. at least one ordering such that
x1 < · · · < xn and that the x indeterminates are precisely the main indeterminates of the
elements of A. The initial of a polynomial pi is the leading coefficient of pi , viewed as a
univariate polynomial in xi . The separant of pi is the polynomial ∂pi /∂xi .
In the following, h denotes either the product of the initials of all the elements of A or
the product of the separants of all the elements of A.
We are concerned by the properties of the ideal A = (A) : h∞ which is the set of all
the polynomials f ∈ R such that, for some nonnegative integer r and some λ1 , . . . , λn ∈ R
we have hr f = λ1 p1 + · · · + λn pn . When h is the product of the initials of the elements
∞ in the literature. When h is the product of the
of A, the ideal A is often denoted (A) : IA
∞.
separants, the ideal A is often denoted (A) : SA
In general, the ideal A may be the trivial ideal R (take A = {x1 , x1 x2 }). We assume
this is not the case.
Denote R0 = K(t1 , . . . , tm )[x1 , . . . , xn ] the polynomial ring obtained by “moving the t
indeterminates in the base field” of R and A0 the ideal (A) : h∞ in the ring R0 . Denote M
the multiplicative family K[t1 , . . . , tm ] \ {0} so that R0 = M −1 R. Denote M/A the image
of M by the canonical ring homomorphism R → R/A. The elements of R0 /A0 , which is
isomorphic to (M/A)−1 (R/A), have the form a/b where a ∈ R/A and b ∈ M/A. In this
section, we prove the following theorem.
Theorem 1.1. An element a ∈ R/A is zero (respectively regular1 ) if and only if every
1
regular = not a zero divisor.
2
element a/b ∈ R0 /A0 is zero (respectively regular).
Proposition 1.2. To prove Theorem 1.1, it is sufficient to prove that every element of M/A
is regular.
Proof. This is a very classical proposition. If every element of M/A is regular then R0 /A0 , is
a subring of the total ring of fractions of R/A [24, chapter IV, paragraph 9]. The proposition
then follows [24, chapter I, paragraph 19, Corollary 1].
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Let us recall the Lasker–Noether theorem [24, chapter IV, Theorems 4 and 6].
Theorem 1.3. (Lasker–Noether theorem)
In a noetherian ring, every ideal is a finite intersection of primary ideals. Every representation of an ideal A as an intersection of primary ideals can be minimized by removing on
the one hand the redundant primary ideals and by grouping on the other hand the primary
ideals whose intersection is itself primary. The so obtained minimal primary decomposition
of A is not uniquely defined. However, the number of its components and the radicals of its
components (the so called “associated prime ideals” of A) are uniquely defined.
All the rings considered in this section are noetherian.
Proposition 1.4. To prove Theorem 1.1, it is sufficient to prove that no associated prime
ideal of A meets M .
Proof. According to [24, chapter IV, paragraph 6, Corollary 3], if M does not meet any
associated prime ideal of A then every element of M/A is regular. Theorem 1.1 then follows
from Proposition 1.2.
Recall the definition of the dimension of an ideal.
Definition 1.5. The dimension dim p of a prime ideal p of a polynomial ring R with
coefficients in a field K is the transcendence degree of the fraction field of R/p over K. The
dimension dim B of an ideal B of R is the maximum of the dimensions of the associated
prime ideals of B.
The rest of this section is completely dedicated to the proof of the following theorem
which admits Theorem 1.1 as a corollary. This reformulation of Theorem 1.1 is often
convenient for writing proofs.
Theorem 1.6. The associated prime ideals of A have dimension m and do not meet M .
In order to apply Macaulay’s unmixedness theorem, one needs to get rid of the saturation
by h. For this, one may use the Rabinowitsch trick [20, section 16.5]. One introduces some
new indeterminate xn+1 and a new polynomial pn+1 = h xn+1 − 1. One denotes A′ the
triangular system of R′ = R[xn+1 ] obtained by adjoining pn+1 to A. One denotes A′ the
ideal (A′ ) of R′ . Consider the two following canonical ring homomorphisms:
φ
π
R −−−→ h−1 R ≃ R′ /(pn+1 ) ←−−− R′ .
3
The isomorphism h−1 R ≃ R′ /(pn+1 ) is classical [7, Exercise 2.2, page 79]: every element
of R corresponds to itself, xn+1 corresponds to h−1 . If B is an ideal of R, one denotes h−1 B
or (φB) the ideal of h−1 R generated by φB. If B′ is an ideal of R′ then πB′ is an ideal of
πR′ = R′ /(pn+1 ).
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Lemma 1.7. The ideal A′ is proper. If q′1 ∩ · · · ∩ q′r is a minimal primary decomposition
of A′ then φ−1 (πq′1 ) ∩ · · · ∩ φ−1 (πq′r ) is a minimal primary decomposition of A.
Proof. We use the notations of extensions and contractions defined in [24, chapter IV,
paragraph 8], w.r.t. the ring homomorphism φ so that (φA) = Ae . The ideal πA′ is equal
to the ideal Ae since both ideals admit a same generating family: A. By [24, chapter IV,
Theorem 15 (a)] we have A = Aec since A = A : h∞ . Therefore, since A is assumed to be
proper, so are Ae and A′ .
Consider now a minimal primary decomposition q′1 ∩ · · · ∩ q′r of A′ . According to [24,
chapter IV, paragraph 5, Remark concerning passage to a residue class ring], πq′1 ∩ · · · ∩ πq′r
is a minimal primary decomposition of πA′ = Ae . Since A = Aec , by [24, chapter IV,
Theorem 15 (b) and a comment just above this theorem], the associated prime ideals of A
do not meet M . By [24, chapter IV, Theorem 17] the intersection φ−1 (πq′1 ) ∩ · · · ∩ φ−1 (πq′r )
is a minimal primary decomposition of A.
Proposition 1.8. To prove Theorem 1.6, it is sufficient to prove that the associated prime
ideals of A′ have dimension m and do not meet M .
Proof. Let p′ be an associated prime ideal of A′ and p = φ−1 (πp′ ) the corresponding
associated prime ideal of A according to Lemma 1.7. Let a be an element of the subring R
of R′ . Then a ∈ p′ if and only if a/1 ∈ πp′ and a/1 ∈ πp′ if and only if a ∈ p. Therefore,
if p′ does not meet M then p does not either and dim p ≥ m. If moreover dim p′ =
m then x1 , . . . , xn must depend algebraically on t1 , . . . , tm modulo p′ hence they depend
algebraically on t1 , . . . , tm modulo p and dim p ≤ m. Combining both inequalities, one
concludes that dim p = m.
One distinguishes two sorts of prime ideals associated to an ideal A: the isolated or
minimal ones and the embedded or imbedded ones. An embedded associated prime ideal
of A is an associated prime of A which contains another associated prime ideal of A. In
the context of polynomial rings, its algebraic variety is included (embedded) in that of
the associated prime ideal that it contains. One thus sees that, at least in the context
of polynomial rings, it is much easier to get informations on the minimal associated prime
ideals (they correspond to the irreducible components of the algebraic variety of the ideal [24,
chapter VII, paragraph 3, Corollary 3 to Hilbert’s Nullstellensatz]) than on the embedded
associated prime ideals, which have no such simple geometric meaning (see however [7,
section 3.8] for a geometric interpretation of embedded primes). In our case, the problem
of the minimal associated prime ideals is easily solved by Lemma 1.10. The problem of the
embedded associated prime ideals is solved by a difficult theorem: Macaulay’s unmixedness
theorem. Recall Krull’s principal ideal theorem [24, chapter VII, Theorem 22].
4
Theorem 1.9. (principal ideal theorem)
If a proper ideal A of a ring R = K[x1 , . . . , xn ] admits a generating family formed of k
elements (1 ≤ k ≤ n) then dim A ≥ n − k.
Let us come back to our study of the ideal A′ of R′ .
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Lemma 1.10. The dimension of A′ is m. Moreover, none of the m–dimensional associated
prime ideal of A′ meets M .
Proof. Consider an associated prime ideal p′ of A′ .
First consider the case of h being the product of the initials of the elements of A. Then
none of these initials belongs to p′ (otherwise p′ , which contains h xn+1 − 1 would also
contain 1). Thus x1 , . . . , xn+1 are algebraically dependent on t1 , . . . , tm over K in R′ /p′
(the polynomials of A′ cannot degenerate at all).
Consider now the case of h being the product of the separants of the elements of A. Let
pℓ = ad xdℓ + · · · + a1 xℓ + a0 be any element of A′ . Since its separant sℓ = d ad xd−1
+ · · · + a1
ℓ
does not belong to p′ (otherwise p′ , which contains h xn+1 − 1 would also contain 1), at
least one of the coefficients ad , . . . , a1 does not belong to it2 . Thus x1 , . . . , xn+1 are algebraically dependent on t1 , . . . , tm over K in R′ /p′ (the polynomials of A′ cannot completely
degenerate).
In both cases, x1 , . . . , xn+1 are algebraically dependent on t1 , . . . , tm over K in R′ /p′ .
One then concludes, first that dim p′ ≤ m hence dim A′ ≤ m, second that if dim p′ = m
then p′ ∩ M = ∅. The ideal A′ admits a basis made of n + 1 elements in a polynomial
ring in n + m + 1 indeterminates. According to the principal ideal theorem, dim A′ ≥ m.
Combining both inequalities, one concludes that dim A′ = m.
Let us recall Macaulay’s unmixedness theorem [24, chapter VII, Theorem 26].
Theorem 1.11. (Macaulay’s unmixedness theorem)
If a proper ideal A of a polynomial ring R = K[x1 , . . . , xn ] admits a basis made of
k elements (1 ≤ k ≤ n) and if dim A = n − k then all its associated prime ideals have
dimension n − k.
The following proposition, combined to Proposition 1.8, concludes the proof of Theorem 1.6 hence that of Theorem 1.1.
Proposition 1.12. The associated prime ideals of A′ have dimension m and do not meet M .
Proof. The ideal A′ admits a basis made of n + 1 elements in a polynomial ring in n + m + 1
indeterminates. According to Lemma 1.10, its dimension is m. According to Macaulay’s
unmixedness theorem, all its associated prime ideals have dimension m. According to
Lemma 1.10 again, none of these prime ideals meets M .
Let us state a few easy corollaries to Theorem 1.1.
Corollary 1.13. The minimal primary decomposition of A is uniquely defined.
2
The characteristic zero hypothesis is used here.
5
Proof. By Theorem 1.6, the ideal A has no embedded associated prime ideal. The corollary
then follows [24, chapter IV, paragraph 5, Theorem 8].
Corollary 1.14. Theorems 1.1 and 1.6 hold if A is replaced by any ideal (A) : S ∞ where
S is any subset of R containing h, provided that (A) : S ∞ is proper.
Proof. The ideal (A) : S ∞ is the intersection of the primary components of (A) : h∞ which
do not meet the multiplicative family generated by S. Since the primary components of
(A) : h∞ do not meet M , the primary components of (A) : S ∞ do not meet M either
and, Theorem 1.6 holds for this ideal also. Theorem 1.1 follows from Theorem 1.6 and
Proposition 1.4.
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Corollary 1.15. Every regular element of R0 /A0 is invertible.
Proof. Still a well known theorem. By Theorem 1.6, the ideal A0 has dimension zero. By [24,
chapter VII, paragraph 7], the associated prime ideals of A0 are maximal. The ideal A0 is
thus contained in finitely many prime ideals. There is a bijection [24, chapter III, Theorem
7] between the ideals of R0 which contain A0 and the ideals of R0 /A0 . This bijection maps
prime ideals to prime ideals [24, chapter III, Theorem 11]. The ring R0 /A0 thus involves
only finitely many prime ideals p1 , . . . , pr which are the associated primes of (0). Assume
a ∈ R0 /A0 is regular. By [24, chapter IV, paragraph 6, Corollary 3], the element a belongs
to none of the ideals p1 , . . . , pr . The ideal generated by a must contain 1 since it would
otherwise have associated prime ideals all different from p1 , . . . , pr and there are no such
prime ideals. Thus there exists some ā ∈ R0 /A0 such that a ā = 1 and a is invertible.
Corollary 1.16. Let 1 ≤ i ≤ n be an index. Denote Ai = {p1 , . . . , pi }. If h is the product
of the initials of A, denote hi the product of the initials of the elements of Ai otherwise,
denote hi the product of the separants of the elements of Ai . Denote Ai = (Ai ) : h∞
i . Let
a ∈ R be any polynomial. If a is regular in Ri /Ai then a is regular in R/A.
Proof. Denote R0,i = K(t1 , . . . , tm )[x1 , . . . , xi ] and A0,i = (Ai ) : h∞
i in R0,i . Assume a is
regular in Ri /Ai . Then, by Theorem 1.1 and Corollary 1.15, there exists some ā ∈ R0,i such
that a ā − 1 ∈ A0,i . Since R0,i ⊂ R0 and A0,i ⊂ A0 , we have a ā − 1 ∈ A0 and a is invertible
in R0 /A0 . By Theorem 1.1 again, a is regular in R/A.
2
Lazard’s lemma
In this section, we keep the notations of section 1 but we restrict ourselves to the case of h
being the product of the separants of the elements of A. The ideal A = (A) : h∞ is often
∞ in the literature. It is assumed to be proper.
denoted (A) : SA
Theorem 2.1. (Lazard’s lemma)
The ideal A is radical.
The minimal prime ideals of A have dimension m and do not meet M .
6
Before proceeding, let us consider the basic case of a system A made of a single polynomial p1 = t1 (x1 − 1)3 (x1 − 2). Then the separant h = t1 (x1 − 1)2 (4 x1 − 7) involves as
a factor the polynomial t1 which does not depend on x1 and the multiple factor (x1 − 1)
of p1 . The ideal A is generated by (x1 − 2) and satisfies Theorem 2.1. Observe that the
theorem would not hold in the case of h being the product of the initials of A only. In that
case, the ideal A, which would be generated by (x1 − 1)3 (x1 − 2), would not be radical.
Proposition 2.2. To prove Theorem 2.1, it is sufficient to prove that A0 is radical.
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Proof. The second statement of Lazard’s lemma follows from Theorem 1.6. Let us assume
A0 is radical. Then R0 /A0 does not involve any nilpotent3 element by [24, chapter IV,
Theorem 10 and Corollary]. Thus R/A does not either by Theorem 1.1 (for if a is a non
zero element of R/A then its image a/1 in R0 /A0 is non zero ; if a power ad of a were zero,
then (a/1)d would be zero too and R0 /A0 would involve nilpotent elements). Therefore, A
is radical and the proposition is proved.
In the rest of this section, we prove that A0 is radical by proving that R0 /A0 is isomorphic
to a direct product of fields. Since a direct product of fields does not involve any nilpotent
element, the ideal A0 is radical and the proof of Lazard’s lemma is complete.
Indeed, if R1 , . . . , Rk are rings then one denotes S = R1 × · · · × Rk their direct product.
Elements of S are tuples with k components. Given any two elements a = (a1 , . . . , ak ) and
b = (b1 , . . . , bk ) of S one defines a + b as (a1 + b1 , . . . , ak + bk ) and a b as (a1 b1 , . . . , ak bk ).
In the ring S, zero is equal to (0, . . . , 0) and one is equal to (1, . . . , 1). If the rings Ri do not
involve any nilpotent element then S does not either. This is the case in particular when
the rings Ri are fields. See [24, chapter III, paragraph 13] for an equivalent formulation
based on direct sums. The following theorem is a generalization of the Chinese Remainder
Theorem. See [24, chapter III, paragraph 13, Theorem 32] or [7, Exercise 2.6, page 79].
Theorem 2.3. (Chinese Remainder Theorem)
If A1 , . . . , Ak are ideals of R such that Ai + Aj = R whenever i 6= j then the ring
R/(A1 ∩ · · · ∩ Ak ) is isomorphic to the direct product (R/A1 ) × · · · × (R/Ak ).
The proposition below concludes the proof of Theorem 2.1. The scheme of its proof is
the original scheme of proof communicated by Daniel Lazard.
Proposition 2.4. The ring R0 /A0 is isomorphic to a direct product of fields.
Proof. The ring R0 /A0 can be constructed incrementally. It is isomorphic to the ring Sn
defined by:
S0 = K(t1 , . . . , tm ), Si = Si−1 [xi ]/(pi ) : s∞
i .
The proof is an induction on n.
The basis n = 0 is trivial.
Assume Sn−1 is a direct product of fields K1 × · · · × Kr . Then Sn is isomorphic to the
direct product of the rings Kj [xn ]/(pn ) : s∞
n for all 1 ≤ j ≤ r. In the formula above one
3
A nilpotent element of a ring R is a nonzero element of R a power of which is zero.
7
assimilates the polynomials pn and sn with their images by the canonical ring homomorphisms, noticing that the image of the separant of pn in each Kj [xn ] is the separant of the
image of pn in this ring.
Therefore, in each Kj [xn ], the ideal (pn ) : s∞
n is generated by the product of the simple
irreducible factors of pn . It is thus the intersection of the maximal ideals mℓ generated
by these factors. According to the Chinese Remainder Theorem, each Kj [xn ]/(pn ) : s∞
n
is isomorphic to the direct product of the fields Kj [xn ]/mℓ . Since direct products are
associative, the ring Sn itself is a direct product of fields.
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Corollary 2.5. Theorem 2.1 holds if A is replaced by any ideal (A) : S ∞ where S is any
subset of R containing the separants of the elements of A, provided that (A) : S ∞ is proper.
Proof. The ideal (A) : S ∞ is the intersection of the primary components of A which do not
meet the multiplicative family generated by S. Since A is radical, its primary components
are prime ideals [24, chapter IV, Theorem 5]. Thus the primary components of (A) : S ∞ are
prime ideals and (A) : S ∞ is radical. The dimension properties shared by all the associated
prime ideals of A also hold for all the associated prime ideals of (A) : S ∞ .
3
Regular chains
We consider the polynomial ring R = K[x1 , . . . , xn , t1 , . . . , tm ]. We assume that the m + n
indeterminates are ordered according to some total ordering O. Let p be any polynomial
of R \ K. The greatest indeterminate w.r.t. O among the indeterminates p depends on is
called the main indeterminate of p. We consider a triangular system A = {p1 , . . . , pn } of R
i.e. a polynomial system whose elements have distinct main indeterminates. Renaming
the indeterminates if necessary, we assume that the main indeterminate of pi is xi for each
1 ≤ i ≤ n. The multiplicative family M , the initials and the separants of the elements of A
are then defined as in section 1.
Fix some 1 ≤ i ≤ n. Denote Ai the system {p1 , . . . , pi }. Denote hi the product of the
initials of the elements of Ai . Denote Ri the ring K[t1 , . . . , tm , x1 , . . . , xi ]. Denote R0,i the
: ∞
ring K(t1 , . . . , tm )[x1 , . . . , xi ]. Denote Ai the ideal (Ai ) : h∞
i of R and A0,i the ideal (Ai ) hi
of R0,i . Denote R0 = R0,n and A = An .
Definition 3.1. The system A is a regular chain if, for each 2 ≤ i ≤ n, the initial of pi is
regular in the ring Ri−1 /Ai−1 . Assume A is a regular chain. Then A is said to be squarefree
if, for each 1 ≤ i ≤ n, the separant of pi is regular in Ri /Ai .
The above definition is not exactly the same as that of [2, Definition 4.1] but they are
strictly equivalent. The difference is that, in [2], the t indeterminates greater than xi would
have been withdrawn from the rings Ri−1 and Ri . This change is not important for the
elements of Ai do not depend on the t indeterminates greater than xi and, by [24, chapter I,
paragraph 16, Theorem 6], if R̄ is a ring, a is one of its elements and x is an indeterminate
over it then a is zero (respectively regular) if and only if it is zero (respectively regular) in
the ring R̄[x]. The following results are corollaries to Theorems 1.1 and 2.1.
8
Corollary 3.2. The system A is a regular chain if, for each 2 ≤ i ≤ n, the initial of pi is
invertible in the ring R0,i−1 /A0,i−1 . Assume A is a regular chain. Then A is squarefree if,
for each 1 ≤ i ≤ n, the separant of pi is invertible in R0,i /A0,i .
Proof. It is an immediate corollary of Theorem 1.1 (enlarging the set of the t indeterminates
with the x indeterminates which are not needed).
Corollary 3.3. Assume A is a squarefree regular chain. Then A is radical. Its minimal
prime ideals have dimension m and do not meet M .
Proof. By Corollary 1.16 and the definition of squarefreeness, the separants of the elements
of A are regular in R/A. Thus they do not lie in any associated prime ideal of A by [24,
∞ the multiplicative family that
chapter IV, paragraph 6, Corollary 3]. Thus, denoting SA
∞ and the proof follows from Corollary 2.5.
they generate, A = A : SA
hal-00137158, version 1 - 22 Mar 2007
3.1
Splittings
In this section, we provide two propositions which permit to justify many algorithms carrying out the “D 5 ” principle for triangular systems [6]. We keep the notations of section 3 and
we assume that A is a regular chain. Let 1 ≤ i ≤ n be an index. Assume that there exists a
factorization pi = b c in (R0,i−1 /A0,i−1 )[xi ] such that 0 < deg(b, xi ), deg(c, xi ) < deg(pi , xi ).
For each 1 ≤ j ≤ n, denote Bj = Aj if j < i otherwise denote Bj = (Aj \{pi })∪{b}. Denote
B = Bn . For each 1 ≤ j ≤ n, denote hb,j the product of the initials of the elements of Bj
: ∞
and Bj the ideal (Bj ) : h∞
b,j of R and B0,j the ideal (Bj ) hb,j of R0,j . Replacing b by c in the
formulas, define C and for each 1 ≤ j ≤ n, define Cj , hc,j , Cj and C0,j Denote B0 = B0,n
and C0 = C0,n .
Proposition 3.4. The triangular sets B and C are regular chains. For each 1 ≤ j ≤ n we
have Aj ⊂ Bj and Aj ⊂ Cj .
Proof. We focus on the set B. The arguments for C are similar. Since 0 < deg(b, xi ) <
deg(pi , xi ), the set B is triangular. For each 1 ≤ j < i we have Aj = Bj thus Aj = Bj and B
is a regular chain up to index i − 1. In the ring R0,i−1 , the initial of pi is the product of the
initials of b and c. Since A is a regular chain, the initial of pi is invertible in R0,i−1 /A0,i−1 .
Therefore, the initial of b is invertible in R0,i−1 /B0,i−1 . By Corollary 3.2, the set B is a
regular chain up to index i and Ai ⊂ Bi . Let i < j ≤ n be an index. We have Aj ⊂ Bj .
Thus the initial of pj , which is invertible in R0,j−1 /A0,j−1 , is also invertible in R0,j−1 /B0,j−1 .
By Corollary 3.2, the set B is a regular chain up to any index and Aj ⊂ Bj .
We have proved that A ⊂ B ∩ C. In general the equality does not hold because of
possible common factors of b and c. In the particular case of squarefree regular chains, b
and c have no common factors and the equality holds, the following proposition shows.
Proposition 3.5. Assume A is squarefree. Then so are B, C and we have A = B ∩ C.
Moreover, the sets of the minimal prime ideals of B and C form a partition of the set of
the minimal prime ideals of A.
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Proof. First we prove that B and C are squarefree regular chains. As in the above proof,
we focus on B. By Proposition 3.4, the set B is a regular chain. Assume A is squarefree.
Let 1 ≤ j < i be an index. Since Aj = Bj , the separant of pj is regular in Rj /Bj and B is a
squarefree regular chain up to index i − 1. Denote si , sb and sc the separants of pi , b and c.
We have si = sb c + sc b. Let us prove that sb is regular in Ri /Bi . Since A is squarefree, si
is invertible in R0,i /A0,i . By Proposition 3.4, for each 1 ≤ j ≤ n we have Aj ⊂ Bj thus si
is also invertible in R0,i /B0,i . Then, using the fact that b ∈ Bi we see that sb c, hence sb ,
is invertible in R0,i /B0,i . By Corollary 3.2, the set B is a squarefree regular chain up to
index i. Let i < j ≤ n be an index. Using again the fact that Aj ⊂ Bj , we see that the
separant of pj is regular in Rj /Bj . Thus B is a squarefree regular chain up to any index.
Similar statements prove that C is a squarefree regular chain.
By Corollary 3.3, the ideals B and C are radical. They are equal to the intersections
of their minimal prime ideals by [24, chapter IV, Theorem 5]. To conclude the proof of
the proposition, it is thus sufficient to prove that the sets of the minimal prime ideals of B
and C form a partition of the set of the minimal prime ideals of A. Denote V , Vb and Vc
the sets of zeros of A0 , B0 and C0 in the algebraic closure of K(t1 , . . . , tm ). Since these
ideals have dimension zero, these sets are finite. The minimal prime ideals of A, B and C
have dimension m and do not meet M . Therefore, by [24, chapter IV, Theorem 15(d)], the
ring homomorphism R → R0 provides a bijection between the minimal prime ideals of A
(respectively B, C) and those of A0 (respectively B0 , C0 ) hence, using [24, chapter VII,
paragraph 3, Corollary 2], a bijection between the minimal prime ideals of A (respectively
B, C) and the elements of V (respectively Vb , Vc ). It is thus sufficient Q
to prove that Vb
and Vc form a partition of V . The cardinal |V | of V is the product nj=1 deg(pj , xj ).
Similar statements hold for Vb and Vc . Since deg(pi , xi ) = deg(b, xi ) + deg(c, xi ) we see
first that |V | = |Vb | + |Vc |. Second, we have Vb ⊂ V and Vc ⊂ V . Third, Vb ∩ Vc is empty for
a common zero of B and C would annihilate si = sb c + sc b which is invertible. Therefore
V = Vb ∪ Vc , the sets Vb and Vc form a partition of V and the proposition is proved.
3.2
The D 5 principle for triangular systems
In this section, we provide the scheme of many algorithms carrying out the “D 5 ” principle
for triangular systems. More efficient algorithms can be found in [13, 12]. See also [22, 21, 5].
The triangular set A is assumed to be a regular chain.
Definition 3.6. For every a ∈ R we define the pseudoremainder of a by A as
prem(a, A) = prem(. . . prem(prem(a, pn , xn ), pn−1 , xn−1 ) . . . , p1 , x1 ).
def
The pseudoremainder algorithm is based on [24, chapter I, paragraph 16, Theorem 9].
It is defined in [10, volume 2, page 407]. The next proposition is proved in [2, Theorem 6.1].
Proposition 3.7. For every a ∈ R we have a ∈ A if and only if prem(a, A) = 0.
The parameter a of algebraic inverse denotes an element of R. The function returns an
inverse of a in R0 /A0 or fails. If it succeeds then a is proved invertible in R0 /A0 hence regular
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hal-00137158, version 1 - 22 Mar 2007
in R/A by Theorem 1.1. The function thus implicitly relies on the equidimensionnality
theorem. If it fails by encountering a zero divisor, it exhibits a nontrivial factorization of
some element pi of A. The exhibited factorization might allow some calling function to
split A as two regular chains by using Proposition 3.4. Observe that the function may fail
even if a is regular in R/A for it checks the regularity of many different elements of R/A.
function algebraic inverse (a, A)
begin
if a ∈ K(t1 , . . . , tm ) then
if a 6= 0 then
1/a
else
the inverse computation fails (inversion of zero)
endif
else
let xi be the main indeterminate of a
(u1 , u2 , u3 ) := extended Euclid (a, pi , xi , A)
if u3 6= 1 then
the inverse computation fails (inversion of a zero divisor): u3 is a factor of pi
else
u1
endif
endif
end
Here is the generalization of the extended Euclidean algorithm called by algebraic inverse.
The main indeterminate of the two polynomials a and b is xi . The polynomials a and b
are viewed as polynomials in (R0,i−1 /A0,i−1 )[xi ]. The function fails or returns a triple
U = (u1 , u2 , u3 ) of elements of (R0,i−1 /A0,i−1 )[xi ] satisfying a Bézout identity in the ring
(R0,i−1 /A0,i−1 )[xi ] i.e. a relation u1 a + u2 b = u3 . A proof that U satisfies a Bézout identity
can be designed by using the two following loop invariants (i.e. properties which hold each
time the loop condition is evaluated). These loop invariants are natural generalizations of
the very classical loop invariants of the basic extended Euclidean algorithm:
• u1 a + u2 b = u3 and v1 a + v2 b = v3 in (R0,i−1 /A0,i−1 )[xi ] ;
• the set of the common divisors of u3 and v3 is equal to the set of the common divisors
of a and b.
Observe that the second invariant is stated without using the word “gcd” which would be
controversial in this context for the ring (R0,i−1 /A0,i−1 )[xi ] is not a UFD. A definition of
the gcd in this context is however provided in [13]. Observe that the function needs to
recognize zero in R0,i−1 /A0,i−1 in order to evaluate the loop condition and to determine the
degree of u3 after the loop execution. This is achieved by Proposition 3.7. The function also
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needs to check the regularity of the leading coefficient of v3 before performing the Euclidean
division. This can be achieved using algebraic inverse.
function extended Euclid (a, b, xi , A)
begin
U := (1, 0, a)
V := (0, 1, b)
while v3 6= 0 do
q := the quotient of the Euclidean division of u3 by v3 in (R0,i−1 /A0,i−1 )[xi ]
T := V
V := U − q V
U := T
done
deg(u3 , xi )
c := the coefficient of xi
in u3
return algebraic inverse (c, A) U
end
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Implantation en Axiom. PhD thesis, Univ. Paris VI, 1999.
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Journal of Symbolic Computation, 28:105–124, 1999.
[3] François Boulier, Daniel Lazard, François Ollivier, and Michel Petitot. Representation for the
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[10] Donald Erwin Knuth. The art of computer programming. Addison–Wesley, 1966. Second
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François Boulier
LIFL, Université Lille I, 59655 Villeneuve d’Ascq, France
[email protected], http://www.lifl.fr/˜boulier
François Lemaire
LIFL, Université Lille I, 59655 Villeneuve d’Ascq, France
[email protected], http://www.lifl.fr/˜lemaire
Marc Moreno Maza
ORCCA, University of Western Ontario, London, N6A 5B7 Canada
[email protected], http://www.csd.uwo.ca/˜moreno
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